Development of a Numerical Model to Describe Steady Streaming in Unsteady Axisymmetric Biological Flows in a Geometry with a Slowly Varying Cross-Section

Laboratory of Computational Engineering Dr. Mark Sawley Laboratory of Fluid Mechanics Prof. Peter Monkewitz - Radboud Nelissen Development of a Numerical Model to Describe Steady Streaming in Unsteady Axisymmetric Biological Flows in a Geometry with a Slowly Varying Cross-Section

Laboratory of Computational Engineering Dr. Mark Sawley Laboratory of Fluid Mechanics Prof. Peter Monkewitz - Radboud Nelissen

Anthony Claude

Abstract

Cerebrospinal fluid moves from the cranial cavities all along the cerebrospinal canal in a pulsatile way. Its flow has been studied here through various geometries. From the basic case of a tube to a model of an idealized anatomical case, velocity profile and flow behaviour were plotted, analyzed and compared. This bi-dimensional study takes into account variation of geometry of the cerebrospinal canal and pulsatile character of the flow. A data base relating variation of internal and external wall for an adult has been realized with Visual Human Server1. This research allows determining Reynolds and Womersley numbers average for a given region of the spinal canal. Quantitative solutions of unsteady analysis through a cylindrical tube and qualitative results concerning steady streaming in geometries with slowly variations of section have been found . Finally comparison were carried out between computational calculations made in 1998 by A. Sarkar and G. Jayaraman2 and Fluent calculations for a similar issue.

1http://visiblehuman.epfl.ch/vhve.php 2Correction to flow rate - pressure drop relation in coronary angioplasty: steady straeming effect Contents

I Basic Fluid Mechanics Applied to Biological Flows 4

II Womersley Number’s Influence on Biological Flows 5

1 Introduction 5

2 Pulsatile flow through a tube 6 2.1 Hypotheses ...... 6 2.2 Velocity’s expression ...... 6 2.3 Results’ Interpretation ...... 9 2.4 RMS-velocity’s expression ...... 11

3 Pulsatile flow through an annular tube 12 3.1 Velocity’s expression ...... 12 3.2 Results’ Interpretation ...... 13

III Numerical Flow Modelling 14

4 Introduction 14

5 Steady state flow - analytical vs. numerical results 15 5.1 Analytical solution ...... 15 5.2 Numerical solution ...... 18 5.3 Results’ interpretation ...... 20

6 CSF flow through an axisymmetric annular model 21 6.1 Generating the mesh ...... 22 6.2 Results’ interpretation ...... 23

7 CSF flow through 2-D models with varying cross-sections 33 7.1 Visual Human Server - Database ...... 33 7.2 Theoritecal part ...... 35 7.3 Axisymmetric geometry with a linear variation of section ...... 37 7.4 Axisymmetric geometry with a half cosinusoidal variation of section . . . . 39 7.5 Axisymmetric geometry with a cosinusoidal variation of geometry . . . . . 40 7.6 Sarkar and Jayaraman case ...... 43

8 Conclusion 46

9 Bibliography 47 Introduction

Biofliudics has gained in importance in recent years, forcing engineers to use fluid me- chanic and apply it to biological situation. Cardiovascular and more generally every human flow circulation problems are very important nowadays. Because of the multiple diseases caused by flow comportment, it is fundamental to understand its characteristics and the rules by which they operate.. Why does an artery get stenosed? How could we optimized drug dispersion in the cerebrospinal canal? What is the influence of a cardiac valve on the blood flow? By answering these questions, we could offer better treatments and solutions to patients.

At the EPFL, some experimental researches are conducted on the spread of drugs ad- ministered directly in the spinal canal (with a catheter). Furthermore, this project will also focus on cerebrospinal fluid flow. Its aim is to evaluate the potential of Fluent solver to calculate accurate solution for oscillatory flow.

With the development of numerical simulation possibilities and the increase of preci- sion and liability of computer programs, research progresses faster. But technology progress is not enough, we still need an engineer to check, analyse and interpret results. That is why good knowledge in fluid mechanics is needed before solving a problem numerically. To what extent is it possible to consider the results obtained as an exact solution? Is it possible to control the precision, and if so, in what way? This project follows this path.

Unfortunately the spinal canal geometry is extremely complex and varies between peo- ple. Considering that and the calculation time needed, it is not appropriate to model precisely the cerebrospinal canal. In order to study flow behaviour of these geometries with varying cross-section, an axisymmetric analytical model will be used. 4

Part I Basic Fluid Mechanics Applied to Biological Flows

As every incompressible fluid, biological one obeys the different conservation principles. Traditionally we consider the conservation of the mass, the momentum and the energy: dρ + ρ∇ · v = 0 (1) dt Considering the incompressibility condition (ρ = cte) we obtain:

∇ · v = 0 (2)

The conservation of the momentum using the Gauss theorem (local form) is ∂ (ρv) + ∇ · (ρvv) = ∇ · σ (3) ∂t Biological fluid flows are unsteady. Thus, they are driven by the gravitational force and the pressure gradient force (cardiac cycle governs this pulsatile flow). Some shear forces are also noticed(caused by the fluid’s viscosity).Many biological flow issue can be discribed analytically. For instance, recirculation caused by a stenosis in an artery. This phenomenon is important in hemodynamic because there is a risk of canal’s obstruction. Considering the cerebrospinal canal, we encounter the same problem. The difference is that the variation of geometry is due to the change of internal and external radius or to the nerve’s insertions and not to deposits.Most of the biological incompressible problems are solved using the Navier-Stokes equations with cylindrical coordinate. Navier-Stokes equation (in cylindrical coordinate) are:

 2  ∂vr ∂vr vθ ∂vr ∂vr vθ • ρ ∂t + vr ∂r + r ∂θ + vz ∂z − r

 2 2 2  ∂p ∂ vr 1 ∂ vr ∂ vr 1 ∂vr 2 ∂vθ vr = − ∂r + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r − r2 ∂θ − r2 + ρfr   ∂vθ ∂vθ vθ ∂vθ ∂vθ vθvr • ρ ∂t + vr ∂r + r ∂θ + vz ∂z + r

 2 2 2  1 ∂p ∂ vθ 1 ∂ vθ ∂ vθ 1 ∂vθ 2 ∂vr vθ = − r ∂θ + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r + r2 ∂θ − r2 + ρfθ   ∂vz ∂vz vθ ∂vz ∂vz • ρ ∂t + vr ∂r + r ∂θ + vz ∂z

 2 2 2  ∂p ∂ vz 1 ∂ vz ∂ vz 1 ∂vz = − ∂z + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r + ρfz 5

Part II Womersley Number’s Influence on Biological Flows

1 Introduction

Before undertaking more complex analysis of the cerebro-spinal canal, a study of an oscillatory fluid flow through a simple geometry is recommanded. The first part of this project is dedicated to an analysis of a flow moving by a pressure gradient (representing the cardiac cycle) in a cylindrical tube.

It is crucial to understand the basic physics of the issue before developing a numeri- cal model. Therefore, the following development can be viewed as a theoretical approach of unsteady flows, particularly, of biological flows.

The first step will be to solve Navier-Stokes equation in order to find an expression for the velocity. Once this is done, the profile shape can be plotted and observed. Finally, a discussion of the physical interpretation and of the parameters’ influence will take place.

The aim here is to define the influence of the Womersley number on a fluid flow and its effect on the velocity profile.

Cerebrospinal fluid - functions and properties The main function of the cerebrospinal fluid is to protect the brain. In fact, by bathing in this fluid, it does not feel effect of the pressure fluctuations. Obviously there are some limits and this protection is efficient as far as we are in a standard situation. Normally, CSF volume and blood volume vary inversely, maintaining the intracranial pressure within normal limits. CSF has also several other important functions: it regulates chemical environment of the central nervous system and it is active in the transportation of biological substances like proteins. It is principally composed by plasma with few proteins and cells and it is considered as a Newtonian flow

g m2 Density ρ [ m3 ] Viscosity µ [ s ] CSF 993 0.0067

Table 1: CSF’s properties 2 PULSATILE FLOW THROUGH A TUBE 6

2 Pulsatile flow through a tube

As a first approximation, the cerebrospinal canal will be modeled as a cylindrical tube. By doing so, the equations are simpler but further from reality. The main objective here is to study the influence of the Womersley number on a fluid flow and on its velocity profile.

Figure 1: One-dimensional fluid flow in a tube

2.1 Hypotheses • Incompressible flow, no external forces

∂P iωt • Pressure grandient is periodic (sinus): ∂z = Ge , with G constant • One-dimensional fluid flow: (0, 0,W ) avec W = W (R, t)

∂ • Axisymmetric flow: ∂θ = 0 D • Boundary condition: W ( 2 , t) = 0

2.2 Velocity’s expression After some simplifications, Navier-Stokes equations become:

∂W ∂P  1 ∂ ∂W  ρ = − + µ (R ) (4) ∂t ∂z R ∂R ∂R ∂P ∂P = = 0 (5) ∂R ∂θ ∂P = Geiωt (6) ∂z To introduce the Womersley number and to make the calculation easier, it is prefer- able to make this problem adimensional:

τ t = ω R = R0r W = W0w 2 PULSATILE FLOW THROUGH A TUBE 7

q ω Womersley number α: α = R0 ν

W0R0 Reynolds number Re: Re = ν

∂W 1 ∂P  1 ∂ ∂W  = − + ν (R ) (7) ∂t ρ ∂z R ∂R ∂R

∂w 2 ∂p ν 1 ∂  ∂w  ωW0 = −W0 + rW0 (8) ∂τ ∂z R0r R0 ∂r ∂r ∂w ∂p ∂2w 1 ∂w α2 = −ReR + + (9) ∂τ 0 ∂z ∂r2 r ∂r

As mentioned aboved, the pressure gradient oscillates. It’s expression is: ∂p = Geiωt = Geiτ (10) ∂z

The consequence of this is that the velocity will be oscillatory too:

w(r, t) =w ˆ(r)eiωt =w ˆ(r)eiτ (11)

To simplify the notation, wˆ(r, t) will be noted w hereafter.

Introducing the equations (10) and (11) in (9):

∂2w eiτ ∂w α2iωeiτ = −ReR Geiτ + eiτ + (12) 0 ∂r2 r ∂r ∂2w 1 ∂w α2iω = −ReR G + + (13) 0 ∂r2 r ∂r

The equation for the velocity w is established by the result of the differential equation √ √ iReGR w(r) = I (αr i)C + K (αr i)C + 0 (14) 0 1 0 2 α2 where I0 = bessel function (first species) order 0 and K0 = bessel function (second species) order 0. However, the equation is not totally defined yet. There are still three constants to find (C1,C2 et G) and the temporal part to add. By examining graphically the K0 and I0 functions, a problem appears on K0’s graph for r = 0: its value is extremely 2 PULSATILE FLOW THROUGH A TUBE 8

important. Physically it is impossible to have an infinite velocity. This implies C2 = 0. Secondly, by using boundary conditions, constant C1 can be found, knowing that there are no velocity on walls r = R0. In other words, we have w(1) = 0. By resolving this equation (Matlab), C1 value is −iReR G C = 0√ (15) 1 2 I0(α R0 i)

By using the flow’s conservation, G is found (suppose that Q is known):

Z 1 Q = 2π wr dr (16) 0 After some transformations: √ 2 I0(αR0 i)α √ G = 2iQ 2 + I0(αR0 i)) (17) ReR02Bα √ R 1 with B = 0 r I0(αr i) dr

Finally, by inserting (17) in (15) and by adding the temporal contribution, the velocity is given by:  √ −iReR G iReR G w(r, τ) = I (αr i) 0√ + 0 eiτ (18) 0 2 2 α I0(αR0 i) α

The next step will be to analyze this expression for different values of α and to give a physical interpretation of the results. 2 PULSATILE FLOW THROUGH A TUBE 9

2.3 Results’ Interpretation Section 2.2 was dedicated to find velocity expression for this problem. After plotting results on Matlab, some observations stand out. In other words, the aim is to explain the influence of the Womersley number on the fluid flow. By looking at the equation (9), it seems that the value of α will influence the unsteady character of the fluid flow. The equation shows that the bigger the Womersley’s number is, the more important will the inertial part be. Profile shapes can be classified in three categories, depending on the α:

• α < 1: the velocity profile has a parabolic shape, but there is a little phase lag (in comparison with the pressure gradient).The maximum amplitude of velocity is centered on the middle of the section (kind of Poiseuille flow)

Figure 2: Womersley number α = 0.1

• α = 1: the profile is close to the first one. However there is no phase lag

Figure 3: Womersley number α = 1 2 PULSATILE FLOW THROUGH A TUBE 10

• α > 1: the flow is phase-shifted in time (much more pronounced than for little α) and the profile is not parabolic anymore. The maximum amplitude of velocity is not centered on the middle anymore. The shape could be separated in two parts: an unsteady boundary layer where viscosity is important and an inviscid core on the center.

Figure 4: Womersley number α = 10

An other interesting point of the analysis is the problem related to the velocity gra- dient on the boundaries. The increase of this gradient will lead to an increase of the shear stress near the wall. By observing the graphic for α > 1 where there is an im- porant velocity gradient, it should also have an important shear stress on the boundaries.

Considering all these figures it is clear that for an oscillatory laminar flow, the Wom- ersley number influences strongly the velocity shape. When the α is small, profile is similar to a Poiseuille flow. The explanation is that pulsations are low enough to allow the parabolic shape to be fully established all along the cycle. Moreover for α = 1 the velocity is exactly in phase with the pressure gradient (it is only right for this value, for others there are delays). When the α is higher, profiles are perturbed because of the high frequency. Velocity profile has not enough time to fully develop. This observation implies a flatter profile and a maximum velocity lower and not centered. 2 PULSATILE FLOW THROUGH A TUBE 11

2.4 RMS-velocity’s expression The RMS-velocity (or Root-Mean-Square-Velocity) is an additional information on the fluid flow. This expression will show us the velocity average during one cycle of time. The equation is given by: s 1 Z 2π RMS(w(r, τ)) = w(r, τ)2 dτ (19) T 0

Considering one cycle and because the problem is non-dimensional, the frequency is 2π. 1 1 calculations are made with T = f = 2π . Then we obtain: s 1 Z 2π RMS(w(r, τ)) = w(r, τ)2 dτ (20) 2π 0

From the equation (15) and (19): w(r, τ) = w(r) eiτ s 1 Z 2π RMS(w(r, τ)) = w(r) e2iτ dτ (21) 2π 0

By making some simplifications, RMS-velocity is finally: 1 RMS(w(r, τ)) = − √ w(r)e2πi (22) 2 π  √  −iReR0√G iReR0G avec w(r) = I0(αr i) 2 + 2 α I0(αR0 i) α

Figure 5: RMS velocity for α = 0.1 and α = 10 3 PULSATILE FLOW THROUGH AN ANNULAR TUBE 12

3 Pulsatile flow through an annular tube

Figure 6: One dimensional fluid flow in a cylindrical tube

3.1 Velocity’s expression Here, results found on section 2 will be re-used. Indeed, differences only concern bound- ary conditions. By re-using velocity equation (14): √ √ iReGR w(r) = I (αr i)C + K (αr i)C + 0 (23) 0 1 0 2 α2 and by re-defining constants C1 and C2 with new boundary conditions, i.e. velocity on the wall equal to zero (w(R0) = 0 and w(R1) = 0):

√ √ −iReR0G −I0(αR1 i)+I0(αR0 i) C1 = 2 √ √ √ √ α (I0(αR0 i)K0(−αR1 i)−K0(−αR0 i)I0(αR1 i) √ √ −iReR0G −K0(−αR0 i)+K0(−αR1 i) C2 = 2 √ √ √ √ α (I0(αR0 i)K0(−αR1 i)−K0(−αR0 i)I0(αR1 i) This case is a bit more complicated because some parameters (Re for example) change depending on the annular tube chosen. Reynolds number is now a function of R1 (because of the non-dimensionalisation, R0 is a reference value). w Re(R ) = characteristic (R − R ) (24) 1 ν 0 1

Flow’s conservation is still used to find G (this value is supposed known), but as explained for Re just before, this constant will now also depend on R1.

Z R0 Q = 2π wr dr (25) R1 After some transformations and per part integrations, G is obtained as an expression depending on R1. Each parameter is now defined. Solving and plotting answers for different cases will allow the understanding flow behaviour (various α through the same annular tube or various tubes for the same α). 3 PULSATILE FLOW THROUGH AN ANNULAR TUBE 13

3.2 Results’ Interpretation

Let us consider first the case illustrated on figures 7 and 8. Here, the situation is R1  R0. The consequence is that the velocity profile shape is quite similar to the classic tube case. Indeed, except for the little maximum velocity shifts, similar behavior is observed: when α is close to one, the shape is parabolic and when α is bigger, the parabola is "flatter". Like quoted for a simple tube, maximum of velocity is lower for higher α. In these cases, shapes are extremely different, and they are representative of their own Womersley number. By observing the velocity profile, it was possible to determine approximately α’s value.

Figure 7: velocity profile for α = 1 Figure 8: velocity profile for α = 10

On figures 9 and 10, internal radius is close to external radius. Visually, shapes seem to be similar, since both are parabolic. However, because of the different Womersley number, frequencies are also different. This implies a delay between these solutions. Here, maximums of velocity are quite similar because ratio R1 is close 1. R0

Figure 9: velocity profile for α = 1 Figure 10: velocity profile for α = 10 14

Part III Numerical Flow Modelling

4 Introduction

Today’s fluid mechanics problems are often extremely complicate to solve analytically. That is why computational engineering takes an important place these last years.

This part is divided in three sections: Analyse of the steady state tube issue will be first done. Exact solution will be compare to some numerical ones to quantify precision obtained with different Fluent configura- tion. However, cerebrospinal flow oscillates. The second section is so dedicated to an unsteady analysis through a tube. Even if the geometry is the same, issue is more complicate. Ve- locity is a time dependent function and boundary conditions are more difficult to define. The purpose is to compare numerical results with calculations made on part 2. Finally tests were undertaken for geometries with slowly variations of section. Objective here is to observe mean magnitude velocity on one cycle to emphasize recirculations. To model geometry close to human anatomy, a database was made using Visual Human Server. It allows observing precisely variations of internal and external diameters and then take into account of this value for the geometry. 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 15

5 Steady state flow - analytical vs. numerical results

Fluent and Matlab can be used to analyze fluid flows. However the process is quite different. When the first one uses numerical path3 to obtain solutions (create a geometry, mesh, solve), the second one uses theoretical and analytical methods: solve equations of motions and Navier Stokes equations by using boundary conditions and then obtain an analytical expression for the velocity (depending on the time and the radius).

5.1 Analytical solution From the results found before, the motion equation has changed. In this first situation, flow is in a steady state and the pressure gradient is constant. Therefore, the problem is simpler. Restart with:

∂W 1 ∂P 1 ∂ ∂W = − + µ( (R )) (26) ∂t ρ ∂z R ∂R ∂R ∂ = 0 (27) ∂t ∂P ∂P = = 0 (28) ∂R ∂θ ∂P = cte = A (29) ∂z

These hypotheses allow to simplify the problem:

1 2 1 ∂ ∂W0w ρW0 A = ν( (R0r )) (30) ρ R0r ∂R0r ∂R0r

2 2 νW0 ∂w ∂ w W0 A = 2 ( + r 2 ) (31) R0r ∂r ∂r 1 ∂w ∂2w ReR A = ( + r ) (32) 0 r ∂r ∂r2

By using the boundary conditions of an annular cylindrical tube (w(R0) = w(R1) = 0) velocity’s expression for a steady state flow is

ReR A R2 − R2 R2log(R ) − R2log(R ) w(r) = 0 r2 − 0 1 log(r) − 0 1 1 0 (33) 4 log(R0) − log(R1) log(R0) − log(R1) with A the constant pressure gradient, given by the flow rate conservation Q = 2π R R0 wr dr R1

3volume finite method 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 16

Using matlab, it is possible to plot various velocity profiles for different R1 and different Re.

Figure 11: velocity profile for an internal radius = 0.2

Figure 12: velocity profile for an internal radius = 0.7

It is interesting to note that the maximum velocity is not centered anymore. By in- creasing the ratio R1 the velocity shape will become similar to a Poiseuille profile whereas R0 by decreasing this same ratio, the maximum velocity will be closer to the internal radius R1. 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 17

The study of the velocity function allows the understanding of the velocity profile shape. As mentioned above, by observing curves we notice that the maximum velocity is not centred when the fluid flows in an annular tube. Its position depends on the ratio R1 R0 (N.b. R1 < 1). R0 What is the reason for that and what is the position of this maximum?

Re-write equation (41):

ReR A R2 − R2 R2log(R ) − R2log(R ) w(r) = 0 r2 − 0 1 log(r) − 0 1 1 0 (34) 4 log(R0) − log(R1) log(R0) − log(R1)

0 The maximum is found by derivation: rmax = r | w (r) = 0

ReR A R2 − R2 1 w0(r) = 0 2r − 0 1 (35) 4 log( R0 ) r R1  2 2  0 ReR0A R0 − R1 1 w (rmax) = 2rmax + = 0 (36) 4 log( R0 ) r R1 v u 2 2 u R0 − R1 rmax(R0,R1) = t (37) 2log( R0 ) R1

The exactitude of this function can be checked by inserting the value plotted on fig- ures 11 and 12 (i.e. R0 = 1 with R1 = 0.2 and R0 = 1 with R1 = 0.7). For the first case (figure 7), rmax = 0.845 and for the second one (figure 8), rmax = 0.546.

An interesting characteristic of this equation is that the radius that gives the maxi- R0−R1 mum velocity is always smaller than 2 .

What would be interesting is to study the comportment for extreme ratios. Indeed, what happens when R0  R1 ? And what about R1 ≈ R0? In the first situation, we should obtain a shape almost similar to an half Poiseuille on every side of the internal radius. However, the difference has to be extremly important to observe this. For example with an external radius of 1km and an internal radius of 1cm, the maximum velocity is on r = 208.4 m. The opposite case (i.e. R0 ≈ R1) gives a solution close to a Poiseuille. For example with an external radius of 1cm and an internal radius of 0.99cm, the maximum velocity is: r ≈ 0.995 m. This value represents the center (similar to the Poiseuille flow) 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 18

5.2 Numerical solution First, an annular geometry similar to the case studied on the fourth part has to be gen- erated and meshed. Naturally, a structured mesh is adapted here. To catch important gradient near to the walls, a boundary layer has to be made. In order to do that a bi- exponant law has been used (similar to the tools "boundary layer" on Gambit, a bit less accurate). The objective here was to obtain, with as smallest possible cell’s number (to reduce the calculation’s time), a convergent solution. However, to ensure mesh quality, tests have been made with other extremely refined mesh and solutions obtain were the same. Solution is independent on the mesh. Finally, inflow and outflow conditions have to be defined. Concerning this study, con- stant velocity inlet and simple outflow is appropriate. Other parameters could have been using (pressure inlet or pressure outlet for example).

Considering the flow rate as a constant and using Loth’s article, a characteristic ve- locity inlet can be defined: wcharacteristic = 0.03537 [m/s] The length of the annular tube needed to obtain a fully developed profile can be deter- mined by the following relation:4

L = 0.06Re(R0 − R1) (38)

Making the assumption that the characteristic Reynolds number is about 20 (Re = wcharac.(R0−R1) ν ) and using an internal radius of R1 = 4.5mm and an external one of R0 = 7.5mm, we obtain L = 3.6mm.

Figure 13: Establishment of a parabolic velocity profile in Poiseuille tube flow

Figure 14: Velocity magnitude of a Poiseuille flow all along the tube L=30mm

4 R0 L corresponds to the length for which boundary layer = 2 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 19

Figure 15: Velocity magnitude in the cen- Figure 16: Velocity magnitude in the cen- ter of the tube - outflow ter of the tube - pressure outlet

Figures 15 and 16 illustrate the length necessary to obtain a fully developed profile. Inlet and outlet zones are problematics. Value export from Fluent had to be in the fully established zone. Entry length could be reduce by imposing an entry function similar to the exact solution obtained analytically. Perturbated outlet zone can be eliminated by imposing an appropriate outflow function (as shown on figure 16).

Figure 17: Inlet and outlet velocity magnitude

In Loth’s article, the Reynolds characteristic number for the CSF is about 200. How does he found this value? Why is it 10 times higher than the one calculated before? First of all, characteristic velocity used before is a constant average velocity. Loth used a velocity profile so magnitudes was not the same. Secondly, Loth calculated Reynolds number each centimeter. Radius vary depending on the region studied. In this part of the project, the tube is constant and characteristic radius have been taken. Finally, mm2 viscosity ν change for each person. Characteristic values are between 1 and 6.5 s . However, both Reynolds imply laminar flows. 5 STEADY STATE FLOW - ANALYTICAL VS. NUMERICAL RESULTS 20

5.3 Results’ interpretation The first result obtained was a graphical comparison between the fully established ve- locity profile found with Fluent and the analitytical solution. Profile shapes are similar

Figure 18: comparison numerical solution / analytical solution even if solutions are not exactly the same (as shown on the right graphic). Difference between them is about 1e−4. This correspond to an error of about 1%. This is acceptable even if a more accurate solution could be hoped for, considering convergence’s criteria imposed (1e−6). It is interesting to analyze whether it is possible to obtain significantly better results by decreasing those criteria. The problem was solved, just changing 1e−6 by 1e−12.

Figure 19: conv.criteria 1e−6-1e−12 1st-2nd order (pressure and momentum discr.)

Those graphics confirm what was expected: naturally more precise results could be ob- tained and logically the scales of these improvements are proportional to the changes of criteria imposed. By proposing 1e−12 instead of 1e−6 the difference between solutions is about 1e−7. Same observation can be made with the order of pressure and momentum discretization for which difference obtain is about 1e−4. To sum up, results obtained with Fluent are satisfying. The errors noted had the same order than the precision asked. 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 21

6 CSF flow through an axisymmetric annular model

Because of the cardiac cycle, most of biological fluid flows are unsteady. As a first ap- proximation we can consider it as a sinusoidal function depending on the time. Obviously human case is not perfect like a sinus but it is accurate enough to help understand the physic of the flow.

Figure 20: cardiac cycle function

The study here concerns an annular tube. Unsteady analysis of this case has been studied on the third part. By remembering equation (11) and using equation (14), pul- satile velocity expression for an annular ube is given by:

 √ √ iReR G w(r, τ) = I (α i)C + K (−α i)C + 0 eiτ (39) 0 2 0 1 α2 with C1 and C2 are constants given by boundary conditions w(R0, t) = w(R1, t) = 0.

√ √ −iReR0G −I0(αR1 i)+I0(αR0 i) C1 = 2 √ √ √ √ α (I0(αR0 i)K0(−αR1 i)−K0(−αR0 i)I0(αR1 i) √ √ −iReR0G −K0(−αR0 i)+K0(−αR1 i) C2 = 2 √ √ √ √ α (I0(αR0 i)K0(−αR1 i)−K0(−αR0 i)I0(αR1 i)

Considering the cerebrospinal canal as an annular cylindrical tube was a good way to test the validity and the precision of Fluent results for an oscillatory fluid flow. 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 22

6.1 Generating the mesh The first step consists in creating an appropriate geometry. Because of the axisymmetry, only a half part of the annular tube has to be made. Rotation turns around x-axis. A refinement near of the boundaries is necessary because of the important pressure gradi- ents expected on these domains. Due to the entry function and the outflow conditions, the mesh will be also refined on these parts to determine easily the regions where the velocity profile will not be fully established.

Because of the simplicity of the geometry, a structured mesh with a small quantity of cells was used (it allowed me to reduce the calculation time). Even if there was the need for a check of its quality to ensure not having convergence problems (aspect ratio for example).

Figure 21: Mesh of the tube

Velocity inlet function w(t) = 0.035sin(2πt) (40)

Mass flow rate (outlet) function 2 2 Q(t) = (0.035sin(2πt))π(R0 − R1)ρ (41) 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 23

6.2 Results’ interpretation To compare solutions and analyze their precision, both of them must have same format. In this project, Matlab was chosen. An adaptation is needed before comparing curves, since nodes are not equidistant (refinement near to boundaries). It is necessary to take into account this parameter because Fluent’s values are given for each node.

Figure 22: Matlab calculated solution

Time Number Residual Pressure-Velocity Momentum Pressure Unsteady Step of step cont−vx − vy coupling discret. discret. formulation A 0.1 40 1e-3 PISO Quick Standard 1st order B 0.05 80 1e-3 PISO Quick Standard 1st order C 0.05 80 1e-5 PISO Quick Standard 1st order D 0.05 80 1e-5 PISO MUSCL Standard 1st order E 0.01 400 1e-5 PISO MUSCL Standard 1st order F 0.05 80 1e-5 PISO Quick Standard 2nd order G 0.05 80 1e-5 PISO MUSCL Standard 2nd order

Table 2: List of cases studied

Time Number Residual Pressure-Velocity Momentum Pressure Unsteady Step of step cont−vx − vy coupling discret. discret. formulation H 0.05 80 1e-5 PISO MUSCL Standard 2nd order

Table 3: Case H: a simple outflow for outlet condition

The aim is to determine the best configuration by comparing all these cases with the analytical solution. The better one will be used for more complex geometries 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 24

Time step influence on solution The smaller time steps are used; the worst estimation of the inlet and outlet sinusoidals is given. This parameter concerns temporal discretization. Time step is for the time discretization comparable to the spacing of the mesh for the spatial discretization. Con- sidering that, it is obvious that a better solution is obtained for little time step. However, it is expensive in calculation time to reduce it.

Figure 23: Error: Matlab - Fluent A Figure 24: Error: Matlab - Fluent B

Figure 25: inlet velocity - time step=.05 Figure 26: inlet velocity - time step=.1 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 25

Differences between these solutions are not evident. To observe important ameliora- tion it may be necessary to reduce even more the time step. Although calculation time becomes really important compared to the precision’s increase given.

Figure 27: Difference between Fluent A and Fluent B 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 26

Value of residuals’ influence on solution By decreasing convergence criteria, precision should increase. Indeed, residual value given corresponds to the maximum error tolerates. The solver will continue to search solution until it arrives to its define value for each parameter (here: continuity, x-velocity and y-velocity).

Figure 28: Error: Matlab - Fluent C Figure 29: Difference Fluent B - Fluent C

A very small variation is observed between solution B (convergence criteria = 1e-3) and solution C (convergence criteria = 1e-5). Theoretically a more significant precision difference was expected here. By observing the convergence history of these cases, an explanation stands out. Residues don’t increase to 1e-5 with the maximum iterations per time step given (and in most of the situations, the problem is not caused by this parameter, the residue cannot increase more). 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 27

Momentum Discretization: Quick/MUSCL influence on solution To be able to solve governing equation, Fluent use a control-volume technique. This means that equations are converting from an integral form to an algebraically one Each volume control (defined by the mesh) has to conserve each quantity.

Nfaces I X ...dAf → ...Af (42) f Z ...dV → ...V (43) V Quick scheme is appropriate for quadrilateral and hexahedral meshes. It is based on a central interpolation of the variable. Quick scheme is accurate and generally used with structured grids aligned with flow direction.

MUSCL scheme (Monotone Upstream-Centred Schemes for Conservation Laws) is a third order scheme. Unlike the Quick scheme, the MUSCL scheme is available for arbi- trary mesh. However, for simple geometry studied here, Quick should be sufficient.

Figure 30: Error: Matlab - Fluent D Figure 31: Difference Fluent C - Fluent D

As expected considering schemes’ properties, difference between these solutions is neg- ligible (1e-6). This parameter should have more importance when the study concerns more complex geometries 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 28

Time discretization order influence on solution Solving unsteady problems needs a spatial and a time discretization. By using a seg- regate solver, the only available choice is using implicit time integration. However, an alternative is possible concerning the order of the time discretization (second or first). Difference between 1st and 2nd order is about 1e-4. This is significant but the main

Figure 32: Error: Matlab - Fluent F Figure 33: Difference Fluent C - Fluent F problem is not here. Indeed, by observing the comparison between analytical solution and F, the error is still of about 10%. Why isn’t it there further difference between these solutions? The reason is quite logical. By increasing the time step of a first order time discretization we should approach the second order approximation. Time step used here are small. Figures 27 and 28 prove it.

The first order accurate temporal discretization is given by

∂φ φn+1 − φn = (44) ∂t ∆t The second order temporal discretization is given by

∂φ 3φn+1 − 4φn + φn−1 = (45) ∂t 2∆t When the first equation take only account into value at the next time level, the second one takes account into the next and the previous time level. Precision could be better. 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 29

Outflow condition influence on solution For most of the cases studied, a mass flow outlet was imposed (dependent on the time). However, remembering steady state analysis, an outflow condition could be efficient here. What is the influence of this function? A negligeable difference is observed (figures 34 and 35). The error is still about 10%. Moreover, solution converged easier with the outflow condition. Considering that, why should this outlet function be imposed?

Figure 34: Error: Matlab - Fluent H Figure 35: Difference Fluent C - Fluent H

First of all, datas exported has been taken in a fully developed region. This could explain the similarity. However, perturbed zones (inlet and outlet) are longer when simple outflow is imposed. To avoid them, the solution is to propose an inlet function = analytical solution and a periodic outlet condition. 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 30

Explanation, propositions to ameliorate solutions Comparing with steady state analysis, results found for unsteady flows are not really convincing. Error is of about 10% for all configurations. What could be the reasons? First are problems compared exactly the same. Is it the same to compare an oscillatory pressure gradient (analytical solution) with a sinusoidal velocity inlet and mass flow rate?

An other explanation could be the numerical approximations made. An example is UDFs’ functions used in Fluent: amplitude proposed for the sinus is given with only two significant decimals. Same approximations are quoted in Matlab code. By avoiding these little mistakes, error should be reduced.

Finally the problem could be due to the mesh. The convergence problem sometimes encountered could confirm this hypothesis. However, convergence was at least assumed for a residue of 1e-4. Error should then be smaller. In order to facilitate convergence, it is possible to vary relaxation’s factors. An other possibility would be to propose a better time step to avoid sinus critical regions (when sinus value is near to zero).

Figure 36: Residuals per time (outlet cond- Figure 37: Residuals per time (outlet cond- tion: outflow = 1) tion: mass flow rate (t)) 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 31

However by observing the velocity average on one sinus cycle, a very small value was found (1e-8). Indeed, for a fully established flow this situation is expected because the inlet velocity is a sinus function. The error is therefore compensated. This is really important because even if the solution is not perfect, Fluent respects flows are imposed, thus, by studying the "real" case, results obtained will not be very precise numerically but correct physically. Another way to check if the velocity profile is fully developed is to observe the radial velocity. It should be (and it is) small, at least as a convergence criteria.

Figure 38: velocity average on one sinus cycle 6 CSF FLOW THROUGH AN AXISYMMETRIC ANNULAR MODEL 32

An other possibility is an error concerning the evaluation of the constant G (mag- nitude of oscillatory pressure gradient). This parameter can be checked using Fluent. Figure 37 represents static pressure all along the tube. Pressure gradient is defined by: ∂P iωt ∂z = Ge . ∂P Using the graph, ∂z ≈ −0.025. Knowing that this graph was plotted for t = 5s and that ω = 2π, G is about -0.029. Theoretical G value is about -0.032.

Figure 39: velocity average on one sinus cycle 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 33

7 CSF flow through 2-D models with varying cross-sections

7.1 Visual Human Server - Database Visible human server (VHS) is a virtual anatomic construction kit. Elaborate at the EPFL, this applet allows observing, isolating, extracting slice and measuring each part of the human anatomy. of the cerebrospinal canal is extremely complicated. In this project, hypotheses are made and supposition is that the variation of geometry is caused only by the contraction and the enlargement of the cross-section. In reality, spinal cavity viewed from the side is also curved. The elaboration of this database needs precision. Values had to be taken each centimetre, all along the spinal canal. Once this is done, we are able to reconstruct human geometry.

Figure 40: Visible Human Server applet 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 34

Figure 41: Hydraulic diameter Figure 42: Hydraulic diameter and cross-section and cross-section area (Loth) area (VHS)

Figure 43: Linear variation of section throug the thoracic 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 35

7.2 Theoritecal part Here, the cerebro-spinal canal is considered as a tube in another one. The difference between those two cylinders is that the external one has a non-constant section. This variation will be first very slow with the aim to make some Fluent tests. Then flows through geometry closer to human anatomy will be studied Mathematically, the situation is almost the same than in the first part, except for the radius. Now, R0 = R0(z) with  R −∞ < z < a  01 R0(z) = R01 (1 − f(z)) a < z < b   R02 b < z < ∞

To guarantee the slowly variation, R0 and R1 are imposed close to each other. To ensure the continuity, f(z) must satisfy R0(1 − f(a)) = R0 and R0(1 − f(b)) = R1

Figure 44: case 1: slow section’s variation for an internal flow

For the non-dimensionalisation, a geometric parameter δ is added, defined by R δ = 0 (46) L Obviously, this is a bidimensional case here and the velocity will have by the same two components: vr and vz (vr  vz). Here the goal is not to solve the governing equations but by observing the Sarkar and Jayaraman’s article, a re-circulation is detected before and after the geometric variation. Even if I’m not exactly in the same case (they studied the influence of a stenosis) We can quite logically assume that it will be the same kind of comportment in my modelisation. The next step will consist in confirming the above by a numerical modelisation and in quantify the results. 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 36

Continuity of f(z) Continuity of f’(z) Continuity of f”(z) Other, notices Linear variation yes no no frequency Half-cosinus yes yes no magnitude = 0.1 frequency Cosinus 1 yes yes yes magnitude = 0.01 frequency Cosinus 2 yes yes yes magnitude = 0.1 frequency Cosinus 3 yes yes yes magnitude = 1

Table 4: List of cases studied

Considering the results found before, all future simulations will be based on configuration G:

Time Number Residual Pressure-Velocity Momentum Pressure Unsteady Step of step cont−vx − vy coupling discret. discret. formulation G 0.05 80 1e-5 PISO MUSCL Standard 2nd order

Table 5: Solver configuration 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 37

7.3 Axisymmetric geometry with a linear variation of section By studying this simple geometry, the goal was to observe the behavior of a pulsatile flow when it encountered a slowly linear variation of section. Obviously, this geometry does not correspond to any physiological case.

Meshing part Mesh and geometry were generated with Gambit. Because of the variation of the section, quad-mesh was not appropriate anymore. To catch variations on the wall, a boundary layer was created. Tri-mesh was imposed for the rest of the tube (Delaunay method) with a particular refinement on the critical regions. Boundary conditions imposed were the same than problems solved before: a sinusoidal velocity inlet and a mass flow rate (sinusoidal too) on the outlet.

Figure 45: Mesh and geometry Figure 46: Focus on the critical region

Solver and solution It is interesting to observe the average velocity magnitude on one cycle of the sinus. Compared to geometries without any variation of section where this value was almost 0, there are here three recirculation’s zones caused by the geometry (convective term on equations).

Plotting mean axial velocity average (and not the magnitude) allows showing the di- rection of the flow. When the geometry "grows up" two of the three recirculations are negative and one is positive (the middle one). When the section decreases, the opposite happens: two of the three recirculations are positive. Thus this configuration satisfies the continuity (illustrations are on the next page). 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 38

Figure 47: Mean velocity magn. Figure 48: Mean velocity magnitude

Figure 49: Mean axial velocity 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 39

7.4 Axisymmetric geometry with a half cosinusoidal variation of sec- tion The following geometry corresponds a bit more to an anatomical case. There is a slowly variation without discontinuity for f’(z). With this example the objective is to understand the influence of this characteristic.

Meshing part

Figure 50: Mesh and geometry Figure 51: Focus on the critical region

Solver and solution The aim is to compare this solution with the one just before. Results are very similar: same order and three recirculation’s zone. The main difference concerns the centre of these regions. With a smoother variation, maximum magnitude is exactly centred.

Figure 52: Mean velocity magn. Figure 53: Mean velocity magnitude 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 40

7.5 Axisymmetric geometry with a cosinusoidal variation of geometry The aim of the following is to compare and observe steady streaming in geometries with a cosinusoidal variation of external radius. The difference between these cases is the ratio frequency δ = magnitude imposed. By simulation, mean velocity magnitude found for each geometry should vary proportionally with δ.

δ = 0.01: Meshing part The mesh was realized with quad-boundary layers on walls. Because of the extremely slow variation of section, mesh could have been completely made by quadrilateral cells.

Figure 54: Mesh and geometry Figure 55: Focus on the critical region

δ = 0.01: Solver and solution

Figure 56: Mean velocity magnitude 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 41

δ = 0.1: Meshing part

Figure 57: Mesh and geometry Figure 58: Focus on the critical region

δ = 0.1: Solver and solution The symmetry of the flow observed here could be explained by the symmetry of the geometry. The only difference concerns the sense of the flow as shown on figure 45.

Figure 59: Mean velocity magnitude Figure 60: Mean axial velocity 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 42

Finally, it was interesting to study an extreme situation in order to observe the flow frequency conduct. The geometry here is based on a cosinus that has a ratio magnitude = 1.

δ = 1: Meshing part

Figure 61: Mesh and geometry Figure 62: Focus on the critical region

δ = 1: Solver and solution This geometry is extreme and the solution found is asymmetric. The reason might be the important region out of the main flow. However, there are also recirculation zones here, but as for the linear case, maximum average velocity of this region is not centred. Deviation is extremely important.

Figure 63: Mean velocity magnitude Figure 64: Mean velocity magnitude 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 43

7.6 Sarkar and Jayaraman case In 1998, A. Sarkar and G. Jayaraman wrote an article intituled "correction flow rate - pressure drop relation in coronary angioplasty: steady streaming effect". They studied the behaviour of fluid flow through stenosed artery (steady streaming, pressure gradient, wall shear stress). This case corresponds to an oscillatory flow through the annularly section of concentric tubes (the external one has a varying diameter). First, they observe on the streamline patterns that the particles in the upstream are moving in the clockwise, unlike those in the downstream that moves in anticlockwise. Then they note the presence of secondary recirculation near to the catheter and stenosis walls.

The purpose here is to compare qualitatively their results to mine, for a similar geometry.

Meshing part Like for enlarging, contracting geometries imply recirculation regions. The mesh has to be made consequently. That’s why a refinement has been made near to the expected critical zones. Quad-boundary layers and tri-mesh generated by Delaunay-method.

Figure 65: Mesh and geometry Figure 66: Focus on the critical region 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 44

Solver and solutions Solver configuration does not change. Only the UDF mass flow rate has to be modified because of the new section of geometry.

Womersley number α = 5 :

Many differences can be observed between these solutions. They observed only two recirculation zones and Fluent, with the same conditions, found three of these. However some reserves could be posed concerning the precision of the streamline they plotted. Indeed, how could a streamline finish on the stenosis wall? This solution seems to be wrong physically. Meshing they used or 1998’s computer performances could be the rea- son for these results. Regarding Fluent solution, continuity of the streamline and symmetry are respected.

Figure 67: Sarkar solution Figure 68: Fluent solution 7 CSF FLOW THROUGH 2-D MODELS WITH VARYING CROSS-SECTIONS 45

Womersley number α = 10 :

Even if results are not exactly similar, it seems to be closer for α = 10. In both cases, there are three recirculation zones, each of these represents almost one third of the radius.

Figure 69: Sarkar solution Figure 70: Fluent solution

Comparison α = 5 with α = 10 :

By comparing the results found with Fluent, an important visual similarity has to be acknowledged. However, as expected, magnitudes are different. In part II, where was observed the behaviour of a flow depending on their Womersley number α, a relation be- tween velocity magnitude and α was found: for the same geometry, velocity magnitude increases when Womersley number decreases. That is what happens here. 8 CONCLUSION 46

8 Conclusion

The importance of Womersley and Reynolds numbers in unsteady flows and particularly in biological flows has been verified through this project. As well in geometry with small cross-section than in geometry with important cross-section, these parameters influence profile shape and velocity magnitude. Their importance in circulation’s issue is manifest. When geometry contains variations of sections, the pulsatile fashion of the flow implies recirculations. Anatomically, this process is to take into account. Effects of this "mixing" can be positive or negative. It could avoid or increase deposits on walls, but could also be problematic near to bifurcations for example.

Even if results found in this study show different solutions between numerical and analyt- ical velocity profile, error seems to more heavily due to the definition of the problem than to Fluent capacity. The error found here is important considering the simple geometry studied (about 10%). However, as mentioned previously, error is compensated on one cycle. In conclusion, the main difficulty in these cases is to define the right issue, with same properties and conditions. Both solutions are certainly correct, but the problem solved must be a bit different.

Future improvement It could be interesting to continue searching the reason of the error found on section 6. Some hypotheses and propositions deserve some numerical tests. The main is imposing the exact velocity solution as entry function. Defining I0 and K0 in an UDF’s function would be the main difficulty.

Future studies could also focus on dispersion through geometry with variations of sec- tions. Has the presence of this second phase any influence on the flow? By considering this second phase as a drug, it could be interesting to observe its dispersion and evolution through an enlargement or a contraction. 9 BIBLIOGRAPHY 47

9 Bibliography

Books • E. Guyon, J.-P. Hulin, L. Petit - Hydrodynamique physique EDP Sciences; 2001

• M.L. Riethmuller - Biological fluid dynamics Rhode Saint Genèse: VKI ; 2003 - Von Karman Institute for Fluid Dynamics

• H. Schlichting, K. Gersten - Boundary-layer theory Berlin: Springer; 2000

• E. Kreyszig - Advanced engineering mathematics New York: Wiley; 1993

• I.L. Ryhming - Dynamique des fluides Deuxième édition; Presses Polytechniques et Universitaires Romandes

• Y.C. Fung - Biomechanics Circulation Springer second edition

Articles • A. Sarkar, G. Jayaraman - Correction to flow rate - pressure drop rela- tion in coronary angioplasty: steady streaming effect Journal of Biomechanic 31, 1998, pp. 781-791

• F. Loth, A. Yardimci, N. Alperin - Hydrodynamic modeling of cere- brospinal fluid motion within the spinal cavity Journal of Biomechanical Engineering, 2001, Vol. 123 pp. 71-79

• C. Loudon, A. Tordesillas - The use of the dimensionless Womersley number to characterize the unsteady nature of internal flow Journal of Theoretical Biology, 1998, Vol. 191 pp. 63-78

• N.B. Wood - Aspect of fluid dynamics applied to the larger arteries Journal of Theoretical Biology, 1999, Vol. 199 pp. 137-161

• K. Rohlf, G. Tenti - The role of the Womersley number in pulsatile blood flow - a theoretical study of the Csson model Journal of Biomechanics 34, 2001, pp.141-148 9 BIBLIOGRAPHY 48

Acknowledgements

Radboud Nelissen, (LMF) For his patience, his advices, his knowledge and his investment in this project

Dr. Mark Sawley, (LIN) For this interesting project

Marc-Antoine Habisreutinger, (LIN) For precious advices he gave me about Fluent

Maxime Pepinster, Sohrab Kehtari, Jannick Fenner and Marcel Vonlanthen For their moral support and their LATEX advices

Sophia Kadri, For her moral support and her english advices